The Asymptotic Analysis for the Singularly Perturbed Subdiffusion Equations on Bounded Domain

Abstract

In this paper, we study the asymptotic properties  of  the singularly perturbed subdiffusion equations in a bounded domain. First, we use the matched asymptotic expansion method to obtain the uniform asymptotic expansion for the solution $u(x,t)$ of the singularly perturbed subdiffusion equation. By asymptotic analysis, we can know that near the boundary configured with non-smooth boundary values, the solution $u(x,t)$ of the singularly perturbed subdiffusion equation has a boundary layer of thickness $\mathcal{O}(\varepsilon)$. By studying the asymptotic properties of the spatial partial derivatives $\partial_xu(x,t)$ and $\partial_{xx}u(x,t)$, we can know that the singularity is mainly concentrated in the boundary layers, and then  the solution $u(x,t)$ changes gently outside the boundary layers. Next, we introduce a new $\mathcal{L}1$-TFPM scheme to solve the singularly perturbed subdiffusion equations numerically.  Some numerical experiments can demonstrate the correctness of the asymptotic analysis results.